Here's a list — not complete but still impressive — of old restaurants in Los Angeles that are still in operation. This is not to be confused with the splendid website run by the proprietor of the one you're reading at the moment. Old L.A. Restaurants covers ones that are no longer in operation.
I have not been to the oldest one on the list of those still serving but I have been to the next two. Cole's and Phillipe the Original both opened in 1908 and both claim to have invented the French Dip sandwich, which is still their specialty. I don't know which one to believe and I doubt there's a way to ever settle that one. I do know that I have a small preference for the sandwiches at Phillipe. Also, the parking. The next oldest one listed that I've been to is one I visit often — the Musso & Frank Grill and then maybe a fourth of the rest. If you're of the area, you will probably want to check off the ones you've patronized, too.